Algorithms for Programmers by Arndt J

By Arndt J

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R − 1) be a 2-dimensional array of data7 . 77) k=0 h=0 For a m-dimensional array ax (x = (x1 , x2 , x3 , . . , xm ), xi ∈ 0, 1, 2, . . , Si ) the m-dimensional Fourier transform ck (k = (k1 , k2 , k3 , . . , km ), ki ∈ 0, 1, 2, . . k ... x1 =0 x2 =0 where z = e± 2 π i/n , n = S1 S2 . . k where S = (S1 − 1, S2 − 1, . . 79) x=0 The inverse transform is again the one with the minus in the exponent of z. 80) x=0 which shows that the 2-dimensional FT can be accomplished by using 1-dimensional FTs to transform first the rows and then the columns8 .

E. FFTs for data sets that do not fit into physical RAM1 . In convolution computations it is straight forward to save the transpositions by using the MFA followed by the TMFA. (The data is assumed to be in memory as row0 , row1 , . . e. 1 (matrixfft convolution algorithm) The matrix FFT convolution algorithm: 1 The naive idea to simply try such an FFT with the virtual memory mechanism will of course, due to the non-locality of FFTs, end in eternal harddisk activity CHAPTER 2. CONVOLUTIONS 41 1.

8 (transposed matrix Fourier algorithm) The (TMFA) for the FFT: transposed matrix Fourier algorithm 1. Transpose the matrix. 2. Apply a (length C) FFT on each column (transposed row). 3. Multiply each matrix element (index r, c) by exp(±2 π i r c/n). 4. Apply a (length R) FFT on each row (transposed column). e. g. in unit strides). In radix 2 (or 2n ) algorithms one even has skips of powers of 2, which is particularly bad on computer systems that use direct mapped cache memory: One piece of cache memory is responsible for caching addresses that lie apart by some power of 2.

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